3.95.27 \(\int \frac {(54-99 x-15 x^2+23 x^3+3 x^4) \log (5)+(-54+18 x^2+2 x^3) \log (5) \log (\frac {3}{x})}{81 x^4-18 x^6+x^8+(162 x^3-36 x^5+2 x^7) \log (\frac {3}{x})+(81 x^2-18 x^4+x^6) \log ^2(\frac {3}{x})} \, dx\)

Optimal. Leaf size=30 \[ -\frac {(6+x) \log (5)}{(-3+x) x (3+x) \left (x+\log \left (\frac {3}{x}\right )\right )} \]

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Rubi [F]  time = 1.36, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (54-99 x-15 x^2+23 x^3+3 x^4\right ) \log (5)+\left (-54+18 x^2+2 x^3\right ) \log (5) \log \left (\frac {3}{x}\right )}{81 x^4-18 x^6+x^8+\left (162 x^3-36 x^5+2 x^7\right ) \log \left (\frac {3}{x}\right )+\left (81 x^2-18 x^4+x^6\right ) \log ^2\left (\frac {3}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((54 - 99*x - 15*x^2 + 23*x^3 + 3*x^4)*Log[5] + (-54 + 18*x^2 + 2*x^3)*Log[5]*Log[3/x])/(81*x^4 - 18*x^6 +
x^8 + (162*x^3 - 36*x^5 + 2*x^7)*Log[3/x] + (81*x^2 - 18*x^4 + x^6)*Log[3/x]^2),x]

[Out]

(Log[5]*Defer[Int][1/((-3 + x)*(x + Log[3/x])^2), x])/3 + (2*Log[5]*Defer[Int][1/(x^2*(x + Log[3/x])^2), x])/3
- (5*Log[5]*Defer[Int][1/(x*(x + Log[3/x])^2), x])/9 + (2*Log[5]*Defer[Int][1/((3 + x)*(x + Log[3/x])^2), x])
/9 + (Log[5]*Defer[Int][1/((-3 + x)^2*(x + Log[3/x])), x])/2 - (2*Log[5]*Defer[Int][1/(x^2*(x + Log[3/x])), x]
)/3 + (Log[5]*Defer[Int][1/((3 + x)^2*(x + Log[3/x])), x])/6

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\log (5) \left (54-99 x-15 x^2+23 x^3+3 x^4+2 \left (-27+9 x^2+x^3\right ) \log \left (\frac {3}{x}\right )\right )}{x^2 \left (9-x^2\right )^2 \left (x+\log \left (\frac {3}{x}\right )\right )^2} \, dx\\ &=\log (5) \int \frac {54-99 x-15 x^2+23 x^3+3 x^4+2 \left (-27+9 x^2+x^3\right ) \log \left (\frac {3}{x}\right )}{x^2 \left (9-x^2\right )^2 \left (x+\log \left (\frac {3}{x}\right )\right )^2} \, dx\\ &=\log (5) \int \left (\frac {-6+5 x+x^2}{x^2 \left (-9+x^2\right ) \left (x+\log \left (\frac {3}{x}\right )\right )^2}+\frac {2 \left (-27+9 x^2+x^3\right )}{x^2 \left (-9+x^2\right )^2 \left (x+\log \left (\frac {3}{x}\right )\right )}\right ) \, dx\\ &=\log (5) \int \frac {-6+5 x+x^2}{x^2 \left (-9+x^2\right ) \left (x+\log \left (\frac {3}{x}\right )\right )^2} \, dx+(2 \log (5)) \int \frac {-27+9 x^2+x^3}{x^2 \left (-9+x^2\right )^2 \left (x+\log \left (\frac {3}{x}\right )\right )} \, dx\\ &=\log (5) \int \left (\frac {1}{3 (-3+x) \left (x+\log \left (\frac {3}{x}\right )\right )^2}+\frac {2}{3 x^2 \left (x+\log \left (\frac {3}{x}\right )\right )^2}-\frac {5}{9 x \left (x+\log \left (\frac {3}{x}\right )\right )^2}+\frac {2}{9 (3+x) \left (x+\log \left (\frac {3}{x}\right )\right )^2}\right ) \, dx+(2 \log (5)) \int \left (\frac {1}{4 (-3+x)^2 \left (x+\log \left (\frac {3}{x}\right )\right )}-\frac {1}{3 x^2 \left (x+\log \left (\frac {3}{x}\right )\right )}+\frac {1}{12 (3+x)^2 \left (x+\log \left (\frac {3}{x}\right )\right )}\right ) \, dx\\ &=\frac {1}{6} \log (5) \int \frac {1}{(3+x)^2 \left (x+\log \left (\frac {3}{x}\right )\right )} \, dx+\frac {1}{9} (2 \log (5)) \int \frac {1}{(3+x) \left (x+\log \left (\frac {3}{x}\right )\right )^2} \, dx+\frac {1}{3} \log (5) \int \frac {1}{(-3+x) \left (x+\log \left (\frac {3}{x}\right )\right )^2} \, dx+\frac {1}{2} \log (5) \int \frac {1}{(-3+x)^2 \left (x+\log \left (\frac {3}{x}\right )\right )} \, dx-\frac {1}{9} (5 \log (5)) \int \frac {1}{x \left (x+\log \left (\frac {3}{x}\right )\right )^2} \, dx+\frac {1}{3} (2 \log (5)) \int \frac {1}{x^2 \left (x+\log \left (\frac {3}{x}\right )\right )^2} \, dx-\frac {1}{3} (2 \log (5)) \int \frac {1}{x^2 \left (x+\log \left (\frac {3}{x}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.62, size = 28, normalized size = 0.93 \begin {gather*} \frac {(-6-x) \log (5)}{x \left (-9+x^2\right ) \left (x+\log \left (\frac {3}{x}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((54 - 99*x - 15*x^2 + 23*x^3 + 3*x^4)*Log[5] + (-54 + 18*x^2 + 2*x^3)*Log[5]*Log[3/x])/(81*x^4 - 18
*x^6 + x^8 + (162*x^3 - 36*x^5 + 2*x^7)*Log[3/x] + (81*x^2 - 18*x^4 + x^6)*Log[3/x]^2),x]

[Out]

((-6 - x)*Log[5])/(x*(-9 + x^2)*(x + Log[3/x]))

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fricas [A]  time = 0.63, size = 32, normalized size = 1.07 \begin {gather*} -\frac {{\left (x + 6\right )} \log \relax (5)}{x^{4} - 9 \, x^{2} + {\left (x^{3} - 9 \, x\right )} \log \left (\frac {3}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+18*x^2-54)*log(5)*log(3/x)+(3*x^4+23*x^3-15*x^2-99*x+54)*log(5))/((x^6-18*x^4+81*x^2)*log(3/
x)^2+(2*x^7-36*x^5+162*x^3)*log(3/x)+x^8-18*x^6+81*x^4),x, algorithm="fricas")

[Out]

-(x + 6)*log(5)/(x^4 - 9*x^2 + (x^3 - 9*x)*log(3/x))

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giac [A]  time = 0.32, size = 46, normalized size = 1.53 \begin {gather*} -\frac {\frac {\log \relax (5)}{x^{3}} + \frac {6 \, \log \relax (5)}{x^{4}}}{\frac {\log \left (\frac {3}{x}\right )}{x} - \frac {9}{x^{2}} - \frac {9 \, \log \left (\frac {3}{x}\right )}{x^{3}} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+18*x^2-54)*log(5)*log(3/x)+(3*x^4+23*x^3-15*x^2-99*x+54)*log(5))/((x^6-18*x^4+81*x^2)*log(3/
x)^2+(2*x^7-36*x^5+162*x^3)*log(3/x)+x^8-18*x^6+81*x^4),x, algorithm="giac")

[Out]

-(log(5)/x^3 + 6*log(5)/x^4)/(log(3/x)/x - 9/x^2 - 9*log(3/x)/x^3 + 1)

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maple [A]  time = 0.13, size = 28, normalized size = 0.93

method	result	size
risch	\(-\frac {\ln \relax (5) \left (x +6\right )}{\left (x^{2}-9\right ) x \left (\ln \left (\frac {3}{x}\right )+x \right )}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^3+18*x^2-54)*ln(5)*ln(3/x)+(3*x^4+23*x^3-15*x^2-99*x+54)*ln(5))/((x^6-18*x^4+81*x^2)*ln(3/x)^2+(2*x^
7-36*x^5+162*x^3)*ln(3/x)+x^8-18*x^6+81*x^4),x,method=_RETURNVERBOSE)

[Out]

-ln(5)*(x+6)/(x^2-9)/x/(ln(3/x)+x)

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maxima [A]  time = 0.48, size = 44, normalized size = 1.47 \begin {gather*} -\frac {x \log \relax (5) + 6 \, \log \relax (5)}{x^{4} + x^{3} \log \relax (3) - 9 \, x^{2} - 9 \, x \log \relax (3) - {\left (x^{3} - 9 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+18*x^2-54)*log(5)*log(3/x)+(3*x^4+23*x^3-15*x^2-99*x+54)*log(5))/((x^6-18*x^4+81*x^2)*log(3/
x)^2+(2*x^7-36*x^5+162*x^3)*log(3/x)+x^8-18*x^6+81*x^4),x, algorithm="maxima")

[Out]

-(x*log(5) + 6*log(5))/(x^4 + x^3*log(3) - 9*x^2 - 9*x*log(3) - (x^3 - 9*x)*log(x))

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mupad [B]  time = 9.26, size = 27, normalized size = 0.90 \begin {gather*} -\frac {\ln \relax (5)\,\left (x+6\right )}{x\,\left (x^2-9\right )\,\left (x+\ln \left (\frac {3}{x}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(5)*(23*x^3 - 15*x^2 - 99*x + 3*x^4 + 54) + log(5)*log(3/x)*(18*x^2 + 2*x^3 - 54))/(log(3/x)*(162*x^3
- 36*x^5 + 2*x^7) + log(3/x)^2*(81*x^2 - 18*x^4 + x^6) + 81*x^4 - 18*x^6 + x^8),x)

[Out]

-(log(5)*(x + 6))/(x*(x^2 - 9)*(x + log(3/x)))

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sympy [A]  time = 0.20, size = 31, normalized size = 1.03 \begin {gather*} \frac {- x \log {\relax (5 )} - 6 \log {\relax (5 )}}{x^{4} - 9 x^{2} + \left (x^{3} - 9 x\right ) \log {\left (\frac {3}{x} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**3+18*x**2-54)*ln(5)*ln(3/x)+(3*x**4+23*x**3-15*x**2-99*x+54)*ln(5))/((x**6-18*x**4+81*x**2)*l
n(3/x)**2+(2*x**7-36*x**5+162*x**3)*ln(3/x)+x**8-18*x**6+81*x**4),x)

[Out]

(-x*log(5) - 6*log(5))/(x**4 - 9*x**2 + (x**3 - 9*x)*log(3/x))

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